| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Standard +0.3 This is a straightforward multi-part question testing standard trapezium rule application and basic logarithm laws. Parts (a)-(b) are routine numerical methods. Part (c) requires recognizing that log₁₀(√x) = ½log₁₀(x) and log₁₀(100x³) = 2 + 3log₁₀(x), then using linearity of integration with the given answer—mechanical manipulation rather than problem-solving. Slightly easier than average due to the scaffolded structure and direct application of rules. |
| Spec | 1.06d Natural logarithm: ln(x) function and properties1.06f Laws of logarithms: addition, subtraction, power rules1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States or uses \(h=3\) | B1 | If conflict between stated and used, award bod if one is correct |
| Attempts \(\frac{"3"}{2}\{\log_{10}2 + \log_{10}14 + 2\times(\ldots\ldots)\}\) | M1 | At least two intermediate terms; bracket structure must be correct or implied; allow \(h=4\) |
| \(= \frac{3}{2}\{\log_{10}2+\log_{10}14+2\times(\log_{10}5+\log_{10}8+\log_{10}11)\} = 10.10\) | A1* | Reaches 10.10 (or 10.1) following at least one correct intermediate line and no incorrect lines giving significantly different answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Increase the number of strips | B1 | Also accept "decrease the width of the strips", "use more intervals" o.e. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int_2^{14} \log_{10}\sqrt{x}\, dx = \frac{1}{2} \times 10.10 = 5.05\) | B1 | awrt 5.05 or allow \(\frac{1}{2}\times(a)\) if slip miscopying 10.10; allow even if answer comes from repeat of trapezium rule |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\log_{10}100x^3 = 2+3\log_{10}x\) | B1 | States or implies this result |
| \(\int_2^{14}\log_{10}100x^3\, dx = [2x]_2^{14} + 3\times10.10 = 54.30\) | M1 A1 | M1: \([ax]_2^{14}+3\times10.10\) or equivalent work for finding area of rectangle; use of trapezium rule again is M0; must see use of (a). A1: 54.30, accept 54.3 |
# Question 3:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| States or uses $h=3$ | B1 | If conflict between stated and used, award bod if one is correct |
| Attempts $\frac{"3"}{2}\{\log_{10}2 + \log_{10}14 + 2\times(\ldots\ldots)\}$ | M1 | At least two intermediate terms; bracket structure must be correct or implied; allow $h=4$ |
| $= \frac{3}{2}\{\log_{10}2+\log_{10}14+2\times(\log_{10}5+\log_{10}8+\log_{10}11)\} = 10.10$ | A1* | Reaches 10.10 (or 10.1) following at least one correct intermediate line and no incorrect lines giving significantly different answer |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Increase the number of strips | B1 | Also accept "decrease the width of the strips", "use more intervals" o.e. |
## Part (c)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int_2^{14} \log_{10}\sqrt{x}\, dx = \frac{1}{2} \times 10.10 = 5.05$ | B1 | awrt 5.05 or allow $\frac{1}{2}\times(a)$ if slip miscopying 10.10; allow even if answer comes from repeat of trapezium rule |
## Part (c)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_{10}100x^3 = 2+3\log_{10}x$ | B1 | States or implies this result |
| $\int_2^{14}\log_{10}100x^3\, dx = [2x]_2^{14} + 3\times10.10 = 54.30$ | M1 A1 | M1: $[ax]_2^{14}+3\times10.10$ or equivalent work for finding area of rectangle; use of trapezium rule again is M0; must see use of (a). A1: 54.30, accept 54.3 |3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{124ee19f-8a49-42df-9f4b-5a1cc2139be9-06_725_668_118_639}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows a sketch of part of the curve with equation $y = \log _ { 10 } x$\\
The region $R$, shown shaded in Figure 1, is bounded by the curve, the line with equation $x = 2$, the $x$-axis and the line with equation $x = 14$
Using the trapezium rule with four strips of equal width,
\begin{enumerate}[label=(\alph*)]
\item show that the area of $R$ is approximately 10.10
\item Explain how the trapezium rule could be used to obtain a more accurate estimate for the area of $R$.
\item Using the answer to part (a) and making your method clear, estimate the value of
\begin{enumerate}[label=(\roman*)]
\item $\quad \int _ { 2 } ^ { 14 } \log _ { 10 } \sqrt { x } \mathrm {~d} x$
\item $\int _ { 2 } ^ { 14 } \log _ { 10 } 100 x ^ { 3 } \mathrm {~d} x$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel P2 2021 Q3 [8]}}